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Measuring Fluorescence to Track a Quantum Emitter's State: a Theory

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Measuring Fluorescence to Track a Quantum Emitter's State: a Theory Measuring Fluorescence to Track a Quantum Emitter's State: A Theory Review Philippe Lewalle,1, 2, ∗ Sreenath K. Manikandan,1, 2 Cyril Elouard,1, 2 and Andrew N. Jordan1, 2, 3 1Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA 2Center for Coherence and Quantum Optics, University of Rochester, Rochester, NY 14627, USA 3Institute for Quantum Studies, Chapman University, Orange, CA 92866, USA (Dated: May 29, 2020) We review the continuous monitoring of a qubit through its spontaneous emission, at an introduc- tory level. Contemporary experiments have been able to collect the fluorescence of an artificial atom in a cavity and transmission line, and then make measurements of that emission to obtain diffusive quantum trajectories in the qubit's state. We give a straightforward theoretical overview of such scenarios, using a framework based on Kraus operators derived from a Bayesian update concept; we apply this flexible framework across common types of measurements including photodetection, homodyne, and heterodyne monitoring, and illustrate its equivalence to the stochastic master equa- tion formalism throughout. Special emphasis is given to homodyne (phase{sensitive) monitoring of fluorescence. The examples we develop are used to illustrate basic methods in quantum trajectories, but also to introduce some more advanced topics of contemporary interest, including the arrow of time in quantum measurement, and trajectories following optimal measurement records derived from a variational principle. The derivations we perform lead directly from the development of a simple model to an understanding of recent experimental results. I. INTRODUCTION [12, 56]. Thus, in contrast with the closed{system quan- tum mechanics described by the Schr¨odingerequation The literature on quantum theory and quantum optics alone, a physical description of the measurement process is replete with works concerning the spontaneous emis- necessarily requires that we consider our primary system sion of atoms, across virtually all of its century{long his- as being open. A \quantum trajectory" arises when a se- tory [1{8]. The generic case, in which the excited{state quence of measurements are made in time, such that we population of an emitter decays exponentially on aver- have a time{series of measurement outcomes, and a cor- age due to the spontaneous emission of a photon, is a responding time{series of inferred quantum states of our paradigmatic phenomenon in quantum optics. More re- primary system, based on that information. The process cently, both the theory [9{41] and experiments [42{46] is necessarily stochastic, as there is randomness present about continuous quantum measurement have received in each successive measurement outcome. considerable attention, and seen rapid progress, reveal- Our emphasis will be on tracking the state of a qubit ing new phenomena and insights into the quantum mea- (a two{level quantum system), through its spontaneous surement process [46{51], and applications to quantum emission; this means that the qubit is coupled to a field control [52{54]. In any such generalized measurement(s) mode, which is its \environment" in this scheme, and [12, 55] of some primary system of interest, there must by interrogating this mode in a variety of ways, we will necessarily be some series of interactions between that be able to infer a corresponding evolution of the qubit's system and its environment, which allows for informa- state. A quantized electromagnetic field mode is repre- tion to flow from the primary system to some meter(s) sented by a quantum harmonic oscillator; we will dis- which record the measurement outcome(s) [56]. The in- cuss the cases where we interrogate the output field by teraction between the system and environment will nec- photodetection (effectively an energy measurement), or 1 essarily disturb the system of interest in a random way, by quadrature measurements (homodyne or heterodyne but inferences about that evolution of the system of in- detection are analoguous to making \position" and/or terest can be drawn as long as our measurement brings us \momentum" measurements of the oscillator). We are arXiv:1908.04720v3 [quant-ph] 28 May 2020 the information that the environment \learned" by inter- motivated in large part by recent experimental work, in acting with the system [57]. Generalized measurements which a superconducting transmon qubit is continuously can be weak (a small amount of information is acquired monitored by homodyne or heterodyne detection of its about the system state, with correspondingly little dis- spontaneous emission, leading to diffusive quantum tra- ruption to its prior behavior) [58{62], or strong (e.g. the system is \collapsed" to an eigenstate of the measure- ment operator by a projective measurement, such that we 1 The quadrature space of the field is effectively the phase space have acquired a lot of information at once and disturbed of the quantum harmonic oscillator describing the field mode in the state by corresponding ramifications in the process) question. In other words, a quadrature is analoguous to the \po- sition" or the \momentum" of a quantum harmonic oscillator, and the product of the noise in orthogonal directions in quadra- ture space is bounded by the Heisenberg uncertainty principle. A reader unfamiliar with a quadrature phase space representation ∗ [email protected] of a field mode may benefit from perusing e.g. Refs. [63, 64]. 2 jectories [24, 33, 65{73]. Following the relevant circuit{ (a) (b) QED experiments, the physical setup we have in mind throughout this work involves a single qubit placed in- side a cavity, such that microwave photons emitted by the qubit via spontaneous emission are coupled into a transmission line leading to a measurement device. The hom.∼phase{sensitive (c) setup is designed such that photons emitted by the qubit via spontaneous emission are transmitted to the detec- Ω tor, but photons in other modes (e.g. to implement some QLA r unitary Rabi rotations on the qubit) are not routed to- LO,θ wards it. Such devices allow for high collection efficiency het.∼phase{preserving of emitted photons, in contrast with situations in which an atom emits into free space. See Fig.1 for an illustra- LO,θ rI tion, and guides to the experimental details can be found balanced QLA LO,θ e.g. in Refs. [67{75]. dyne rQ We will proceed by splitting our manuscript into two main parts. In the first part, we describe a qubit open to p (d) ^y a decay channel and subsequent measurements from sev- 1 − η a` eral different perspectives. We carry out the formal treat- a^y any ideal ment of an unmonitored decay channel, using first the p y detector typical quantum{mechanical analysis in Sec. IIA, and η a^s then by introducing the corresponding master equation in Sec. IIB. We transition towards diffusive trajectories in vac. Sec. III, introducing the Kraus operators that will serve as our primary tool, along with the Stochastic Master FIG. 1. We show schematics representing the possible ex- Equation (SME). We examine the cases of heterodyne or perimental situations we consider. Historically, treatments homodyne detection in detail, in sectionsIV andV, re- of spontaneous emission have often focused on the average spectively. We first discuss ideal measurements in which dynamics of emitters, without any monitoring of individual all information is collected, and then describe inefficient emitters and their individual emission events, represented in measurements in which some information is collected and (a). Our emphasis here is on devices in which the emission is some is lost in sec.VB. Each of these steps is repre- captured in a cavity / transmission line, and routed to a de- sented graphically in Fig.1. In the second part, we use tector, such as (b) a photodetector, or (c) a homodyne or het- erodyne setup which measures one or both of the signal field’s the examples developed in the first part as a springboard quadratures. Optically, the single{photon signal is mixed to introduce certain concepts and methods of interest in with a strong local oscillator (LO) on a 50/50 beamsplitter for the current research literature. For example, we are able quadrature detection, leading to a readout of one (homodyne) to use these examples to introduce ideas related to the or both (heterodyne) quadratures of the field. Contemporary arrow of time in quantum trajectories in Sec.VIA, and circuit{QED experiments using microwave photons typically discuss \optimal paths" (OPs) [32, 34, 47, 50, 66, 76] perform these measurements using a quantum{limited ampli- in Secs.VIB andVIC, which are quantum trajectories fier (QLA) built from Joesphson junctions (see e.g. [28, 77{ which connect given states according to an extremal{ 79]). The measurement axis in the quadrature phase space is probability readout, derived according to a variational determined by the relative phase θ between the signal and LO principle. Summary, outlook, and further discussion are / amplifier pump tone. Relevant experiments often include a included in Sec. VII. drive characterized by the Rabi frequency Ω. In (d) we illus- trate a simple model for measurement inefficiency, in which an unbalanced beamsplitter splits the ideal signal into a mea- sured portion with probability η 2 [0; 1] and a lost portion II. UN{MONITORED DECAY (wherea ^y denotes a photon creation operator). See Sec.VB for further details on this last point. We review the case of a single qubit whose fluorescence goes unmonitored in two parts. First we review the stan- dard treatment of Weisskopf and Wigner [3]; next we “artificial atoms" now used in many experiments. The introduce an equivalent master equation description of state of a qubit can be represented as living in the Bloch the system [55, 57, 80]. sphere; we will generically parameterize our single{qubit A qubit is any two{level quantum system; density matrix with Bloch coordinates according to mathematically{speaking this means that it is de- scribed like a spin{ 1 .
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